In the left and central panels of figure REF  we plot the limits fixed by the Auger observations on the SHDM mass \(M_X\)  and the coupling SHDM-SM \(\alpha _{X\Theta }\) , in the case of direct coupling (left panel), and the coupling of the hidden DS gauge interaction \(\alpha _{X}\)  in the case of the instanton-induced decay (central panel). Requiring that the SHDM density today fits the observed DM density, the limits on \(M_X\)  and \(\tau _X\)  can be rewritten in terms of the cosmological parameters characterising the generation mechanism of SHDM. In the right panel of figure REF  we plot the limits coming from Auger data on the age of the universe at the end of inflation \(H_{inf}^{-1}\)  and the efficiency of the reheating phase \(\epsilon =(\Gamma _\phi /H_{inf})^{1/2}\) , being \(\Gamma _\phi =g_\phi ^2 M_\phi /8\pi \)  the inflaton (of mass \(M_\phi \) ) decaying amplitude (see [1]}, [2]} and references therein).
